Recognisability for algebras of infinite trees

Abstract

a b s t r a c tWe develop an algebraic language theory for languages of infinite trees. We define a class of algebras called ω-hyperclones and we show that a language of infinite trees is regular if, and only if, it is recognised by a finitary path-continuous ω-hyperclone.Instead of using finite automata to develop the theory of regular languages, one can also employ semigroup theory. By now this approach has a long tradition and there exists an extended structure theory connecting varieties of languages with finite semigroups. This theory is particularly effective if one is interested in characterising subclasses of the class of all regular languages. For instance, the only known decidable characterisation of the class of first-order definable languages is based on semigroup theory. Naturally, there have been attempts to generalise this theory to other notions of regularity. For languages of ω-words, such a generalisation has largely been achieved. A detailed account can be found in the book of Pin and Perrin [8]. There also have been several contributions to an algebraic theory for languages of finite trees [11,13,12,4,9,6,5,10,3,1,7]. But the resulting theory is still fragmentary with several competing approaches and formalisations. Our own work has been influenced in particular by the following two articles: Ésik and Weil [6] have developed an approach using preclones, while Bojańczyk and Walukiewicz [3] use forest algebras. As far as the algebraic setting is concerned, the formalisation in the present article most closely resembles the work on clones by Ésik [4].So far, an algebraic theory for languages of infinite trees is still missing. The main obstacle is the lack of appropriate combinatorial tools, like Ramseyan factorisation theorems for infinite trees. In particular, a purely combinatorial proof that every nonempty regular language of infinite trees contains a regular tree is still missing. There is recent work of Bojańczyk and Idziaszek [2] on characterisation results for classes of infinite trees that manages to circumvent these problems by a technical trick: since every regular language of infinite trees is determined by the regular trees it contains, it is sufficient to consider only regular trees.This paper provides a first step in the development of an algebraic theory for recognisability of classes of infinite trees.Inspired by the work of Ésik and Weil on preclones, we define suitable algebras of infinite trees called ω-hyperclones.We can show that every regular language is recognised by some homomorphism into such a (finitary, path-continuous) ω-hyperclone. The proof is performed in two steps. First, we define a special class of ω-hyperclones called path-hyperclones that directly correspond to tree automata. The problem with path-hyperclones is that their definition is not axiomatic, but syntactic. That * Corresponding address:

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